The exact information cost of randomized measurements in quantum metrology
Abstract
Randomized measurements provide state-independent readout for multiparameter quantum sensing. Although 3-designs retain a constant fraction of the quantum Fisher information (QFI) for every pure state, we show that 2-designs need not. In dimension d, complete mutually unbiased bases (MUBs) retain exactly 1/(d+1) of the QFI in the worst state and direction, attaining the universal lower bound for 2-designs built from orthonormal bases; symmetric informationally complete (SIC) measurements retain exactly (1−1/√(d+1))/d. The loss comes from aligned blind spots. Relative to a basis, pure-state changes split into population and phase parts of equal dimension, but probabilities respond only to the former. Mutual unbiasedness turns a population change for one basis into a phase change for every other. The worst case already occurs for a qubit encoded in two levels. Independent random bases instead scatter their blind spots. In sufficiently large dimensions, five Haar-random bases retain, with high probability, a dimension-independent QFI fraction for every pure state and direction. Random Clifford measurements on any number of qubits retain exactly one third. Thus blind-spot geometry, not second-order symmetry alone, controls worst-case metrological performance.
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BibTeX
@misc{li2026exact,
title = {{The exact information cost of randomized measurements in quantum metrology}},
author = {Wanqing Li and Jiaxin Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00599},
eprint = {2611.00599},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00599},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The exact information cost of randomized measurements in quantum metrology
AU - Wanqing Li
AU - Jiaxin Liu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00599
UR - https://arxiv.org/abs/2611.00599
AB - Randomized measurements provide state-independent readout for multiparameter quantum sensing. Although 3-designs retain a constant fraction of the quantum Fisher information (QFI) for every pure state, we show that 2-designs need not. In dimension d, complete mutually unbiased bases (MUBs) retain exactly 1/(d+1) of the QFI in the worst state and direction, attaining the universal lower bound for 2-designs built from orthonormal bases; symmetric informationally complete (SIC) measurements retain exactly (1−1/√(d+1))/d. The loss comes from aligned blind spots. Relative to a basis, pure-state changes split into population and phase parts of equal dimension, but probabilities respond only to the former. Mutual unbiasedness turns a population change for one basis into a phase change for every other. The worst case already occurs for a qubit encoded in two levels. Independent random bases instead scatter their blind spots. In sufficiently large dimensions, five Haar-random bases retain, with high probability, a dimension-independent QFI fraction for every pure state and direction. Random Clifford measurements on any number of qubits retain exactly one third. Thus blind-spot geometry, not second-order symmetry alone, controls worst-case metrological performance.
ER -